And what a challenge the magazine's editor, John Kampfner, has set his new columnist. "I should be funnier than John Pilger," says Bremner. "For obvious reasons, that's not a brief John Kampfner can publicly agree to. But that's the joke between us."

Raising more laughs than the rather hectoring Pilger may not be too arduous a task. But it is exactly what the Statesman needs if it is to fulfil Kampfner's ambition of becoming a "treat", known for being amusing and entertaining, as well as intellectually stimulating. The quality of argument in the NS is not in doubt; the number of readers who turn to it for light relief is less sure.

Humour and the left, as Bremner concedes, have not always been the most constant of bedfellows. It's too easy for left-wing comedians to descend into railery - one thinks of Ben Elton in the 1980s - or even to excise humour from their routines completely. This is a pitfall that Kampfner will be keen for Bremner to avoid, although he has not always managed to do so on television. One critic accused him of often being "so clever he forgets to be funny".

Bremner laughs when I mention this. "The watchword is the importance of not being earnest, of not taking it too seriously," he says. "That is a trap I tend to fall into. The approach with the weekly column should be the same as at Bremner, Bird and Fortune - it should be driven by a sense of curiosity on one hand and a sense of humour on the other; pieces in it that make you laugh, but also pieces that really surprise or interest you. And a lightness of touch."

This balance is a difficult one, especially for a left-wing satirist such as Bremner (he is happy to be identified as of the left - "That's OK, everyone's on the left now," he says). And he now admits that it was lacking in Trust Me, I'm A Prime Minister, a much-hyped, one-off special on Tony Blair by Bremner and the two Johns which went out 18 months ago.

"Yeah, we made a mistake with that," he says. "If we'd had another week on that, it would have been a different programme, and a better programme. I think it was too earnest a piece. We saw this plethora of quotes from Tony Blair about Mrs Thatcher, where he said she had lost her sense of right and wrong, and that she was talking only to those closest to her, all of which matched Blair perfectly." The way the quotes were used on screen, however, came across as overly didactic. "They were very stark, white on black. It was quite ponderous. We should have sifted out the stuff that was too serious and too editorialising, and injected more humour into it. Maybe that's the process of editing I need to do with this column - write the thing and then gradually reintroduce the humour."

Bremner then makes a very telling defence of the programme. "If you weren't told that we were funny," he says, "but you were told it was an analysis of Blair from a particular perspective, I think it held up very well."

It has been a journey across the political spectrum for 44-year-old Bremner, from voting Conservative in 1979 to becoming one of Tony Blair's most bitter critics. Why, I ask, did he vote for Mrs Thatcher? "I had an Edinburgh, middle-class childhood and a public school education," he says.

All the same, I say. He points out that he went to Wellington. "That was a Conservative madrassa," he says. "It was a military school with a door marked 'Sandhurst'. For a wet-behind-the-ears 18-year-old, it wasn't surprising. But it didn't take me long to realise that something had gone terribly wrong. And I can remember in 1987 having this despair when it turned out that Thatcher got back in."

By this point, Bremner had his own BBC2 series, Now - Something Else, to which he had graduated after contributing to Spitting Image and Week Ending. He was still known then as a general impressionist, the "new Mike Yarwood", whose speciality was, if anything, in sports commentators (he had a hit in 1985 with N-N-Nineteen Not Out, a spoof based on a Paul Hardcastle number in which Bremner ridiculed the poor performance of the England cricket team). "It originally was a wider thing," he says, "but the change came because of the way television developed.

"When I was growing up, there were just the three channels, so as a nation we all sat down to the same meal at the end of the day. Now there's been this explosion - BBC radio alone has seven channels. So I was drawn increasingly to what happened in the news, perhaps because those programmes were the only ones I was beginning to watch. I would hear the news voices - the Clintons, the Majors, the Clarkes, Bushes and Heseltines."

Bremner watches Newsnight, is an irregular reader of The Independent and The Guardian, and listens to the Today programme "a lot". "It coincides with breakfast, which is very important for the children [he and his second wife, the artist Tessa Campbell Fraser, have two daughters aged four and two]. It's the one meal in the day when we sit down as a family."

From the early 1990s onwards, Bremner's shows, which had moved to Channel 4 and featured his name in various permutations, also contained slots from the veteran satirists John Bird and John Fortune. Their contribution was more fully recognised in 1997 when the title Bremner, Bird and Fortune was adopted.

"I went to the Bird and Fortune madrassa," says Bremner. "I was trained in satire camps in the fine arts of ridicule and in whatever the comic equivalent of suicide bombings is. They radicalised me, and they still do." Bremner describes himself as moving "further and further left", finding himself, in the paradox resulting from the creation of New Labour, most at home with the Liberal Democrats. "I'm happy with that," he says, "because the alternative is too terrible to contemplate."

Keeping his politics in check, or at least making sure he is not too preachy, will be the issue for his column. For it to succeed, the increasingly serious side of Bremner will have to be kept in check by the humorous side which brought him to the public notice in the first place.

Which other columnists does he admire? "Politically," he begins, although he mentions none that do not fall into that category, "John Kampfner, Robin Cook, Matthew D'Ancona in The Sunday Telegraph. Robert Fisk is always interesting. Gore Vidal - there's a kind of Olympian superiority in the way that he writes now. He's almost sending himself up as a parody of himself, and he'd be the first to point that out. There's a lot of Americans - Seymour Hersh, Charles Glass. What I'm interested in is what gets your curiosity going, because there's so much that's just there to fill space."

He tells a story about Jeremy Clarkson. "I have a producer friend whose party trick is to play theme tunes from television on the piano. Clarkson heard him once and really enjoyed it. Then, a few weeks later, he wrote in his column about how terrible these people were who have party tricks. He bumped into the producer afterwards, who said: 'You were really up for that. Why did you say what you did?' And Clarkson allegedly said: 'I had a column to write.' I hope I don't fall into that trap."

Bremner's aim will be to take the most successful aspects of his show - his character impersonations - to his Statesman column. "Time will tell whether that can transcribed onto the page," he says.

For someone who has been at the pinnacle of his profession for so long, he seems remarkably unsure about his new column, which is, after all, in a publication that reaches only a fraction of the audience his television work does.

"You're worrying me now," he says, when I ask him about the difficulty of sustaining a regular slot. "Now I'm instantly nervous about the demands of doing a weekly column." This doubt crops up several times in the conversation. "You're gradually talking me out of this. Just as well I didn't talk to you before I agreed to do it." And later: "I'm not doing it any more!"

It is often said of impressionists that there is a void where their own personalities ought to be, and that the only voices they cannot do are their own. Bremner certainly speaks quietly, and in a voice far less distinctive than those that he employs in his impressions. He is insecure about his status and his achievements. He has a degree from London University in French and German, and has translated into English two operas - Weill's Der Silbersee and Bizet's Carmen - yet describes himself as not "sufficiently rigorous intellectually" to merit the name of "satirist".

"I feel fraudulent using that word," he says. "What the Johns do is satire, what Chris Morris does is satire. But I think what I do is a kind of 'satire light'. In a more intellectually rigorous age, I wouldn't be talked about as a satirist at all. I would just be a topical comedian. People may say that what I do is very clever, but it's not really at all. It's not Swift."

Even his status as a comedian is a cause of doubt. Referring to "what he does" at one point, he adds: "Whether that's comedy or not." He thought that appearing on the panel show Mock The Week would be a form of relaxation. "But actually it just gave me a whole new set of anxieties. In one round, for example, you've got five subjects and you have to think of four lines on each. So you're having to generate 20 lines with all the usual caveats - is this funny? etc - just for one round of an eight-round programme." Bremner admires comedians who can produce brilliant lines off the top of their heads, such as Ross Noble and Eddie Izzard. "I admire the people who can fly like that. I think I just run down the runway very fast, flapping my wings."

Like many stars of the small screen, Bremner has his own company (Vera Productions). Unlike others, however, he has no interest in being a media player himself. "That business aspect of the media, that Charlotte Street world, the advertising and programme executives, it leaves me absolutely cold. I'm supposed to be the director of a television company, but I've only ever seen that company as a vehicle for making the kind of programmes we wanted to make, getting our ideas on the screen."

His disdain for business extends to his politics. The ever-proliferating Private Finance Initiatives are a regular target of his scorn. When he says: "For Blair, it's not about politics, it's about marketing", he hardly needs to add that he's "got more and more distaste for marketing". "Maybe that's why I'm moving further and further to the left," he says. Other aspects of the commercialisation of British society bother him. "The celebritocracy's another thing. People famous for being famous. Are we evolving into a generation that doesn't analyse, that isn't interested in politics?"

As a celebrity himself, he found himself the object of keen tabloid interest a few years ago, after the break-up of his first marriage. Bremner was often pictured looking rather sheepish while out and about escorting his girlfriends, who included the television presenter Penny Smith and Jemima Goldsmith's friend, Zoe Appleyard.

"I was very uncomfortable with all that," he recalls. "It tied in with a time when I felt very unhappy in general. I remember driving home one evening while they were reviewing the papers on the radio. One of the articles was about me separating from my wife. It's a weird thing to listen to a news report about the break-up of your marriage. Very upsetting."

Does that experience inform the way in which he deals with material about the private lives of public figures?

"Yes, it does actually. I'm not particularly comfortable doing that kind of personal stuff unless I feel that the offence is aggravated - say, in the Blunkett case, for example."

What Bremner is comfortable doing is talking about the Prime Minister. The conversation repeatedly turns, unprompted, to the man who once joked over a game of tennis: "How does Lord Bremner sound?" when the impressionist warned that he would have to be as hard on him as he was on John Major. (The tennis playing days are long over.) "He has such vast reservoirs of confidence," says Bremner. "I don't know if he stands in front of a mirror saying: 'You are great, you are great, you are great.' But he is a consummate actor and communicator. It's when you look behind the reality that it begins to fall apart."

Bremner marvels at Blair's ability to "get away with it". His explanation? "It's rather like an illusionist, a David Blaine. You know it's a trick, but you want to believe it."

But Bremner is not only no longer a believer, he has even tired of having to portray Blair on screen. "He's so self-righteous, so priggish, that I don't find him a particularly attractive character to do. Whereas I enjoy someone like Tony Benn or George Galloway." They are proper "characters", he says, with real passions.

Bremner has other ambitions beyond the continuing success of Bremner, Bird and Fortune: to spend more time with his family; to be involved in a modern interpretation of Voltaire's Candide. "All the satirical techniques are there," he says. "Understatement, irony, burlesque, caricatures, false argument."

For now, the immediate challenge is his column in the New Statesman. "In terms of a sharp-eyed media junkie, political columnist, I would say I'm painfully fraudulent," he says. "I just do what I do. I remember someone once saying about John Peel that he had an exaggerated sense of his own unimportance. And during the course of this interview I'm rapidly developing a sense of that."

But people do like his work very much, I say, trying to reassure him. "I just get more nervous as I get older," he says. "There is a moment before I start writing a column when I think: 'What right have you got to be doing this?' " And with that, Bremner is off. He has a private performance to do that night and is worried that his material is not quite right. He worries a lot, this serious man who seems more comfortable in other skins than his own.